Vector Bundles - Vector Bundle, Frame Bundle, Connection, Tangent Bundle, Canonical Bundle, Line Bundle, Vector-Valued Differential Form (Paperback)

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 29. Chapters: Vector bundle, Frame bundle, Connection, Tangent bundle, Canonical bundle, Line bundle, Vector-valued differential form, Ample line bundle, Banach bundle, Beauville-Laszlo theorem, Cotangent bundle, Clifford bundle, Symbol of a differential operator, Normal bundle, Tautological bundle, Lie algebra bundle, Splitting principle, Spinor bundle, Tautological line bundle, Adjoint bundle, Horrocks-Mumford bundle, Birkhoff-Grothendieck theorem, Vector bundles on algebraic curves, Iitaka dimension, Dual bundle, Tractor bundle, Holomorphic vector bundle, Horrocks bundle, Seshadri constant, Inverse bundle, Tango bundle, Exterior bundle, Tensor bundle, Algebraic vector bundle. Excerpt: In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X. The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space V such that V(x) = V for all x in X: in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X V over X. Such vector bundles are said to be trivial. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a manifold we attach the tangent space to the manifold at that point. Tangent bundles are not, in general, trivial bundles: for example, the tangent bundle of the (two dimensional) sphere i...